2017 Indonesia MO Problems/Problem 1
Problem
is a parallelogram. is a line passing . Prove that the distance from to is either the sum or the difference of the distance from to , and the distance from to .
Solution
In order to prove that the distance from to is either the sum or the difference of the distance from to , and the distance from to , we will use casework to show that the statement is true for all scenarios.
Case 1: passes through or
If passes through , then the distance from to is zero. The distance from to is the same as the distance from to because is parallel to . That means the distance from to is the sum of the distance from from to and from to . By using similar steps, if passes through , the distance from to is the sum of the distance from from to and from to .
Case 2: passes through
By SSS Congruency, . Since the area of the two triangles is the same, the distance from to equals the distance from to . Because the distance from to is zero, the distance from to is the difference of the distance from from to and from to .
Case 3: passes through or
Let be the intersection of lines and , and let be points on such that . Also, let , , and , making .
By Alternate Interior Angles Theorem and Vertical Angle Theorem, . Thus, by AA Similarity, .
Using similar triangle ratios, we have and . Thus, we have , so the distance from to is the difference between the distance from to and the distance from to if passes through . By symmetry, we can also show that the distance from to is the difference between the distance from to and the distance from to if passes through .
Case 4: does not intersect the parallelogram at any other points
Let be the intersection of lines and , and let be points on such that . Also, let , , and , making .
By Alternate Interior Angles Theorem, . Thus, by AA Similarity, .
Using similar triangle ratios, we have and . Thus, we have , so the distance from to is the sum of the distance from to and the distance from to if passes through .
In all of the cases, the sum or difference of the distance from to and the distance from to is equal to the distance from to .
See Also
2017 Indonesia MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 2 |
All Indonesia MO Problems and Solutions |